Hyperbolic embeddedness and extension-convergence theorems of \(J\)-holomorphic curves (Q1024204)

This paper deals with Kobayashi hyperbolicity for almost complex manifolds. The analogue of the Kobayashi semi-distance is defined using \(J\)-holomorphic curves and hyperbolicity means that it is a true distance. In the paper under review several results that are classical in the complex case are generalized to this setting. \S 2 deals with hyperbolic embeddings, which are defined as usual (see. Def. 2.1). Denote by \(\mathcal {O}_J(\Delta,M)\) the set of \(J\)-holomorphic discs in the almost complex manifolds \((M,J)\). As in the complex case, a relatively compact almost complex submanifold \(M\) of an almost complex manifold \((N,J)\) is hyperbolically embedded iff \(\mathcal{O}_J(\Delta, M)\) is relatively compact in \(\mathcal{O}_J(\Delta, N)\). Other equivalent conditions are given in terms of length functions (see Thm. 1) and of limit \(J\)-complex lines (see Def. 2.5 and Thm 2). Next in \S 3 the stability of hyperbolically embedded submanifolds is considered under a small perturbation of the almost complex structure. Let \((M,J)\) be a compact almost complex manifold and let \(D \subset M\) be a \(J\)-holomorphic hypersurface. If \((D,J)\) is hyperbolic and \(M\setminus D\) is hyperbolically embedded, then the same holds for almost complex structures close to \(J\) in the \(\mathcal{C}^\infty\) topology (Thm. 3). \S\S 4-5 deal with extension of \(J\)-holomorphic maps from the pointed disc. Results of \textit{P. Kiernan} [Value-Distrib. Theory, Proc. Tulane Univ. Progr. Value-Distrib. Theory complex Analysis related Topics differ. Geom., Part A, 97-107 (1974; Zbl 0292.32018)] and \textit{J. Noguchi} [Invent. Math. 93, No.1, 15--34 (1988; Zbl 0651.32012)] are generalized to the almost complex setting. The proofs are completely different.